On 26 June 1968, as much of Europe was busy rebelling against authority and fighting for free love, Pope Paul VI made a dramatic announcement that put the Roman Catholic church back in the headlines for reasons other than its stance on women, abortion or contraception.

Bones discovered in a Roman cemetery in the Vatican, he declared, had been identified "in a way we believe to be convincing" as those of Saint Peter, the Christian martyr who is traditionally held to have been the first pope and died 1,950 years ago.

On Sunday, fragments of the bones are to be displayed in public as part of celebrations to mark the end of the Year of Faith, an initiative launched by Pope Benedict XVI, who resigned this year.

The fragments, contained in an urn usually kept in a private papal chapel, will be presented for public veneration in St Peter's Square at a mass celebrated by Pope Francis.

The decision to exhibit is controversial. No pontiff has ever said the bones are without doubt those of Saint Peter, and some within archaeological circles are fairly sure they are not.

The battle over the bones, which pits a rigorous Jesuit archaeologist against a pioneering female epigraphist, is one of the strangest stories to have come out of the Vatican during the 20th century and may also be one of the least dignified.

But, speaking on Monday, Monsignor Rino Fisichella said he had no qualms about thrusting the relics back into the spotlight. "We did not want to, and have no intention, of opening up any argument," said Fisichella who, in a carefully worded article for the semi-official Vatican newspaper L'Osservatore Romano last week, described the relics as those "recognised by tradition" as Saint Peter's.

"We believe … faith, the people of God, has always believed these to be the relics of the apostle Peter, and we continue to venerate them in this way and give them the honour they deserve," he said.

Fisichella, president of the pontifical council for the promotion of the new evangelisation, also said "the symbolic value" of the bones – their "underlying theological value" – was hugely important. Regardless of what scientific testing might reveal, he said, Christians would venerate the remains and pray at the tomb of Saint Peter.

The story of how the bones came to be proclaimed Peter's dates back to 1939, when Pope Pius XII ordered an excavation of an area below St Peter's basilica thought to contain his tomb. The digging, overseen by a German monsignor, Ludwig Kaas, lasted 11 years and led, in 1950, to a stunning papal radio broadcast announcing "the tomb of the prince of the apostles" had been found.

But the pope was forced to admit his team had been unable to prove with certainty the bones were Peter's.

Years later, Margherita Guarducci, an archaeologist and the first woman to lead Vatican excavations, began to question the original findings. She noted graffiti near the tomb reading Petr eni, which she believed was an abbreviation of Petros enesti, the Greek for "Peter is here".

She was told Kaas had been collecting bones out of concern that they were not being properly looked after, and putting them in boxes in a Vatican storeroom. Having located some bones she thought were the most interesting, she convinced Pope Paul VI to commission tests on them. These revealed, among other things, that they belonged to a robust man who died approximately in his 60s. To the outrage of Antonio Ferrua, the Jesuit father who had been the chief archaeologist on the initial excavation, Guarducci told the pope he should say the bones were believed to be Saint Peter's. And, to the disquiet of Ferrua and some other Vatican experts, he did just that. Kaas, Ferrua and Guarducci have all since died.

In his book The Vatican Diaries, longtime observer John Thavis calls the affair "an embarrassment" for the church. "The supposed bones of Saint Peter had been surreptitiously dug up by a meddling monsignor when the archaeologists weren't looking; then they were thrown into a box and forgotten for more than a decade; then they were rediscovered by accident and became the focus of a feud between church experts," he writes.

"The whole affair did not inspire confidence in the Vatican's ability to exhume its own history, and it is little wonder that none of it is mentioned in the Vatican guidebooks." The Vatican, however, hopes the bones' moment has finally come. During its Year of Faith, which began in October 2012, 8.5 million pilgrims had prayed at St Peter's tomb, Fisichella said, and it seemed only fitting that the year should be rounded off with "a unique moment".

"For the first time, the relics of the apostle will be displayed for the veneration of believers," he said. "Peter was called by the Lord to confirm his brothers in faith. Around the successor of Peter, but almost in the physical presence of the first of the apostles – to whom, with Paul, we owe the foundation of this church – we will be called to profess our faith once more with conviction and strength."

• This article was amended on 10 January 2014 to make clear that the relics had never been "hidden" as stated in the original article; the novelty of the event was that the relics were being displayed publicly for the first time for veneration in a public ceremony. While the particular relics exhibited had previously been kept in an urn in a private chapel, pilgrims visiting St Peter's tomb beneath the basilica have been able to see others since the area was opened after excavations.